Has it been theoretically proven or dis-proven that a machine can generally think and communicate on its own?
We have not been able to create a truly intelligent AI yet, according to your definition. So we have no real life proof-of-concept that shows that it actually works.

But based on the current research, there is no known property of the human brain that cannot be modeled in software/hardware. We do not understand the human brain enough yet - and most likely lack the computational power and required storage - to create an artificial copy. But as long as we do not identify a property of the human brain that cannot be translated into a piece of software/hardware, it is most likely possible. I honestly cannot even imagine such a property within the bounds of physics that would create such a barrier.

Of course this is no formal proof, but without a clearly identified show-stopper there is no reason to believe that it is not possible to create a thinking machine.